Tucson, 5 June 2002 IODC-IWA2 1
Optical Design of Laser Beam Shaping Systems
David L. Shealy University of Alabama at Birmingham
Department of Physics, 1530 3rd Avenue South, CH310Birmingham, AL 35294-1170 USA
Tucson, 5 June 2002 IODC-IWA2 2
Outline of Presentation
• Overview of history and current practices• Geometrical methods for design• Applications:
• Two-plano-aspheric lens system• Two-mirror system with no central
obscuration• Three-element GRIN system
Tucson, 5 June 2002 IODC-IWA2 3
Historical Background• Frieden, Appl. Opt. 4.11, 1400-1403, 1965: “Lossless
conversion of a plane wave to a plane wave of uniform irradiance.”
• Kreuzer, US patent 3,476,463, 1969: “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equi-phase surface.”
• Rhodes & Shealy, Appl. Opt. 19, 3545-3553, 1980:“Refractive optical systems for irradiance redistribution of collimated radiation – their design and analysis.”
Tucson, 5 June 2002 IODC-IWA2 4
Contemporary Beam Shaping*•Process of redistributing the irradiance and phase
•Two functional categories:•Field Mapping•Beam Integrators
*Laser Beam Shaping: Theory and Techniques, F.M. Dickey & S.C. Holswade,eds., Mercel Dekker, 2000;Laser Beam Shaping, F.M. Dickey & S.C. Holswade,eds., Proc. SPIE 4095, 2000;Laser Beam Shaping, F.M. Dickey, S.C. Holswade, D.L. Shealy, eds., Proc. SPIE 4443, 2001.
Tucson, 5 June 2002 IODC-IWA2 5
Field Mapping Beam Shaper
Tucson, 5 June 2002 IODC-IWA2 6
Beam Integrators
D
d
F f
S
Tucson, 5 June 2002 IODC-IWA2 7
Physical versus Geometrical Optics0 02 2
f r Yπβ
λ=
λ = wavelength, r0 = waist or radius of input beam, Y0= half-width of the desired output dimensionf = focal length of the focusing optic, or the working distance from the
optical system to the target plane
Beam Shaping Guidelines:β < 4, Beam shaping will not produce acceptable results4 < β < 32, Diffraction effects are significantβ > 32, Geometrical optics methods should be adequate
Tucson, 5 June 2002 IODC-IWA2 8
Selected Chapter Titles:
•“Mathematical and Physical Theory of Lossless Beam Shaping,” L.A. Romero, F.M Dickey.
•“Gaussian Beam Shaping: Diffraction Theory and Design,” F.M. Dickey, S.C. Holswade.
•“Geometrical Methods,” D.L. Shealy
•“Optimization-based Techniques for Laser Shaping Optics,” N.C. Evans, D.L. Shealy.
•“Beam Shaping with Diffractive Diffusers,” D.R. Brown.
•“Multi-aperture Beam Integration Systems,” D.M. Brown, F.M. Dickey, L.S. Weichman.
•“Current Technology of Beam Profile Measurements,” C.B. Roundy.
Tucson, 5 June 2002 IODC-IWA2 9
Overview of Geometrical Methods
• Geometrical optics intensity law:( ) 0I∇ =ai
1 1 2 2I dA I dA=
Source
1 1I dA
2 2I dA
• Constant optical path length condition:0( ) ( )rOPL OPL=
Tucson, 5 June 2002 IODC-IWA2 10
Optical Design of Laser Beam Shaping Systems: Differential Equations vs. Global Optimization
• Geometrical methods leads to equations of the optical surfaces:
• Global Optimization with discrete & continuous variables:– Beam shaping merit function
Tucson, 5 June 2002 IODC-IWA2 11
Applications of Geometrical Methods
• Two plano-aspheric lens system for shaping rotationally symmetric Gaussian beam.
• Two mirror system with no central obscuration for shaping elliptical Gaussian beam.
• Three-element GRIN system for shaping rotationally symmetric Gaussian beam.
Tucson, 5 June 2002 IODC-IWA2 12
Two Lens Beam Shaping SystemJiang, Ph.D. Dissertation, UAB, 1993
Energy Balance:
Ray Trace Equation:
Constant OPL:
Tucson, 5 June 2002 IODC-IWA2 13
Input and Output Beam Profiles
Tucson, 5 June 2002 IODC-IWA2 14
Beam Shaping Applications
In a holographic projection processing system featured on 10 January 1999 issue of Applied Optics , a two-lens beam shaping optic increased the quality of micro-optical arrays.
Tucson, 5 June 2002 IODC-IWA2 15
J.A. Hoffnagle & C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. 39.30, 5488-5499, 2000.
Tucson, 5 June 2002 IODC-IWA2 16
Two Mirror Beam Shaping SystemShealy & Chao, Proc SPIE 4443-03, 2001
Input beam has 1-to-3 beam waist in x-to-y directions:Parameters:
0 0const.out X YI A A= =
2 2
0 0
( , ) exp 2 exp 2inx yI x yx y
= − −
Tucson, 5 June 2002 IODC-IWA2 17
Sag of First Mirror
2 20
10; 5L h
l L h
= =
= +
00 0 max 0
00 max 0
max 0 max
1; ;2
3 ; 32
2
rr x x r
ry y r
X r Y
= = =
= =
= =
Parameters:
Tucson, 5 June 2002 IODC-IWA2 18
Optical Performance Analysis*
• ZEMAX was used.
• Elliptical Gaussian input beam profile was modeled as user-defined surface.
• Two mirror surfaces were also modeled as user-defined surfaces.
*Shealy and Chao, Proc. SPIE 4443, 24-35, 2001: “Design and Analysis of an elliptical Gaussian laser beam shaping system.”
Tucson, 5 June 2002 IODC-IWA2 19
Relative IlluminationInput Beam
Output Beam
Tucson, 5 June 2002 IODC-IWA2 20
First Mirror Surface Analysis
Tucson, 5 June 2002 IODC-IWA2 21
Second Mirror Surface Analysis
Tucson, 5 June 2002 IODC-IWA2 22
Tolerance Analysis• In A, the first mirror was decentered 10% of its diameter about x-axis.• In B, both mirrors were decentered by 2.5% of their diameter about x- and y-axes and tilted about the three axes by 0.25 degrees.
Tucson, 5 June 2002 IODC-IWA2 23
Conclusions for 2 Mirror System
• Designed a two-mirror system with no central obscuration for shaping a 3:1 elliptical Gaussian beam into a uniform output beam.
• ZEMAX used to do optical performance analysis: – First mirror has strong aspherical component along
direction of smaller waist (120µm for 6mm diameter mirror).
– Output beam profile is stable for decentering of less than 2.5% of mirror diameter and 0.25 degrees about coordinate axis.
Tucson, 5 June 2002 IODC-IWA2 24
3-Element GRIN Beam Shaping SystemN. C. Evans, D. L. Shealy, Proc. SPIE 4095, pp. 27-39, 2000.
• Can a spherical-surface GRIN shaping system be designed using catalog GRIN materials?
• System would have practical applications.• Problem is well suited for Genetic Algorithms:
– discrete parameters • Number of lens elements • GRIN catalog number
– Continuous parameters: radii, thickness
Tucson, 5 June 2002 IODC-IWA2 25
In itia lize G A by random ly p ick ing new ind iv iduals
E va luate M erit Function for each ind iv idual in generation
P erform genetic operation s (rep roduction , m utation s, cross-overs); p roduce new generation
Is th e popu lation stagnant? (M icro-G A check)
K eep best ind iv idual and rep lace the rem ainder w ith random ly-selected ind iv iduals
End
Y
N
Y
N
T erm ination criterion reached?
Tucson, 5 June 2002 IODC-IWA2 26
Merit Function Used in GA Optimization
( )
( )
22
Target N1Diameter Collimation
Uniformityout out
1 1
exp exp 1 cos ( )
1 1( )
NQ
ii
N N
i ki k
s R RM MM
MI R I R
N N
γ=
= =
− − − − = =
−
∏
∑ ∑
Rtarget = Output Beam RadiusRN = Marginal Ray Height on Output Planeγi = Angle ith Ray Make with the Optical AxisQ and s = Convergence Constants
Tucson, 5 June 2002 IODC-IWA2 27
Parameters Optimized for Free-form GA-designed GRIN ProblemParameter Description Type Limits[i]
1 Number of Elements Discrete 1-4 (integer)2 Radius of curvature of left surface of Element 1 Continuous -100 to 100 3 Radius of curvature of right surface of Element 1 Continuous -100 to 1004 Thickness of Element 1 Continuous 1 to 10 5 Distance between Element 1 and Element 2 Continuous 1 to 10 6 GRIN glass type for Element 1 Discrete 1-6 (integer)7 Positive or Negative GRIN for Element 1 Discrete 0 or 18 Radius of curvature of left surface of Element 2 Continuous -100 to 100 9 Radius of curvature of right surface of Element 2 Continuous -100 to 100 10 Thickness of Element 2 Continuous 1 to 10 11 Distance between Element 2 and Element 3 Continuous 1 to 10 12 GRIN glass type for Element 2 Discrete 1-6 (integer)13 Positive or Negative GRIN for Element 2 Discrete 0 or 114 Radius of curvature of left surface of Element 3 Continuous -100 to 100 15 Radius of curvature of right surface of Element 3 Continuous -100 to 100 16 Thickness of Element 3 Continuous 1 to 10 17 Distance between Element 3 and Element 4 Continuous 1 to 10 18 GRIN glass type for Element 3 Discrete 1-6 (integer)19 Positive or Negative GRIN for Element 3 Discrete 0 or 120 Radius of curvature of left surface of Element 4 Continuous -100 to 100 21 Radius of curvature of right surface of Element 4 Continuous -100 to 100 22 Thickness of Element 4 Continuous 1 to 10 23 Distance between Element 4 and Surface 10 (a dummy surface) Continuous 1 to 10 24 GRIN glass type for Element 4 Discrete 1-6 (integer)25 Positive or Negative GRIN for Element 4 Discrete 0 or 126 Distance from Surface 10 (a dummy surface) to the Output Plane Continuous 1 to 100
Tucson, 5 June 2002 IODC-IWA2 28
Determining when a Solution is Found
Tucson, 5 June 2002 IODC-IWA2 29
3-Element GRIN Shaping System
Element 1
Element 2Element 3
Tucson, 5 June 2002 IODC-IWA2 30
3-Element GRIN Shaping System•Average evaluation time for a generation: 7.80s
•Total execution time: 26.8 hrs
•Integrating Output Profile over Output Surface yields 21.9 units; integrating Input Profile over Input Surface yields 21.7 units
Tucson, 5 June 2002 IODC-IWA2 31
Summary and Conclusions• Geometrical methods for design of laser beam
shaping systems uses:– Conservation of energy within a bundle of rays,– Constant optical path length condition.
• Numerical and analytical techniques used to design a 2-plano-aspherical lens system and a 2-mirror system with no central obscuration.
• Laser beam shaping merit function used with genetic algorithms to design a 3-element GRIN system with spherical surface lenses.